Structural composition law for prime-power matrices

Construct a composition law f⋆f on the divisor-indexed matrices fM(p^e)f such that the product of fM(p^e)f and fM(p^{e'})f yields fM(p^{e+e'})f, and in particular establish whether fM(p^e)f can be derived from repeated applications of fM(p)f.

Background

For coprime integers, the paper establishes the Kronecker-product identity fM(k_1k_2)=M(k_1)⊗M(k_2)f. Section 5.3 asks whether an analogous construction exists for powers of a single prime, where the matrix dimensions change from fe+1f and fe'+1f to fe+e'+1f. The natural candidate, the symmetric product of fM(p)f, is shown not to be conjugate to fM(p2)f, leaving the desired prime-power composition structure unresolved.

References

It is tempting to ask if there could be such results when k = pe is a power of a prime p, with the Kronecker product replaced by a law ⋆ such that the product M(pe) ⋆ M(pe′) of M(pe), of size e + 1, and indexed by Dpe, times M(pe′), of size e′ + 1, is the matrix M(pe+e′) of size e + e′ + 1 indexed by Dpe+e′. In particular, one would like to have M(p)⋆e = M(pe).

— Number of partitions of modular integers (with an Appendix by P. Deligne)  (2502.19523 - Broadhurst et al., 26 Feb 2025) in Section 5.3, pages 16–17